Stochastic control problems with terminal constraints have been extensively studied in the literature. Optimal control problems under stochastic target constraints have been studied in Bouchard, Elie and Imbert [1]} using the geometric dynamic programming principle proposed in Soner and Touzi [2]} . In Föllmer and Leukert [3]}, the authors introduce the notion of quantile hedging to relax almost-sure constraints into probability constraints. In Yong and Zhou [4]} Chapter 3, necessary optimality conditions are proved in the form of a system of forward/backward stochastic differential equations. More recently the problem with constraints on the law of the process has been studied in Pfeiffer [5]} and in Pfeiffer, Tan and Zhou [6]}. In these works, the authors prove that the problem can be reduced to a "standard" problem (without terminal constraint) by adding a term involving \( \lambda ^* h\)  —in the case where the constraint has the form \(\mathbb {E}\left[h(X_T) \right] \le 0\) — to the final cost for some optimal Lagrange multiplier \(\lambda ^*\) . A dual problem over the Lagrange multipliers associated to the constraints is exhibited using abstract duality results. In Pfeiffer, Tan and Zhou [6]}, the authors provide necessary and sufficient optimality conditions for problems with multiple equality and inequality expectation constraints with much less restrictions on the data than we do and in a path dependent framework. However [6]} needs to assume some controllability condition (Assumption \(3.1.ii\) ) and works with a compact control set. In our framework, the corresponding controllability condition would be to assume a priori that there exist some control \((\alpha ^1, \alpha ^2)\)  such that \(\mathbb {E}( h(X_T^{\alpha ^1,\alpha ^2} )) <0\) . In our analysis, this condition is a consequence of the requirements we have on the data.
